Parks in Paris' vincinity: Bois de Vincennes and, a bit wilder, the Bois de Boulogne. Even though large and not in Paris center, these are both city parks, and you should not expect anything wild there.
Closest “real” forest to Paris: Fontainebleau forest. This one has a few deers, but you should not expect to see any (French Wikipédia says less than one per 100 ha…). Boars and badgers are more present, but badgers much decrease their activity during winter. Fontainebleau has birds, but I don't know how many of them are visible in December — I'm not a bird connoisseur, sorry!
Then, pretty much the whole of metropolitan France is accessible by train. If it's for a one-day visit, go to Burgundy, which you can go deep inside in less than 2 hours by TGV (express train network). Small stations like Montbard would allow you to go in the woods by feet, or to rent a bike at the local tourist office. Walking routes are easy to get at the Office du Tourisme.
There, you may expect all the classical European fauna: deers, foxes, boars, rabbits, hares, hawks, herons, hedgehogs… and cows and snails (local specialties ;) ). But with their usual visibility in wild places, that is: very little chance you encounter any.
You could get some really nice sights in the Pyrénées, in places such as Orlu, that also have wolves in semiwild environments… but in the mountains, in December, animals don't spend their time wandering around! Moreover, getting there will start to get long. You could do it with a night train from Paris to Toulouse, then a TER from Toulouse to Ax-les-Thermes (2 hours), but it's really not worth it in this season.
In the end, I can only say that December is not a good time to observe wild animals in Europe (I guess someone could give a list proving the opposite but, as a non-professional-but-nature-loving-hiker, I can only say I haven't seen anything exciting beyond boars and foxes in winter, and they really came as surprises).